# Metrics in the sphere of a C*-module

Esteban Andruchow; Alejandro Varela

Open Mathematics (2007)

- Volume: 5, Issue: 4, page 639-653
- ISSN: 2391-5455

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topEsteban Andruchow, and Alejandro Varela. "Metrics in the sphere of a C*-module." Open Mathematics 5.4 (2007): 639-653. <http://eudml.org/doc/269531>.

@article{EstebanAndruchow2007,

abstract = {Given a unital C*-algebra \[\mathcal \{A\}\]
and a right C*-module \[\mathcal \{X\}\]
over \[\mathcal \{A\}\]
, we consider the problem of finding short smooth curves in the sphere \[\mathcal \{S\}\_\mathcal \{X\} \]
= x ∈ \[\mathcal \{X\}\]
: 〈x, x〉 = 1. Curves in \[\mathcal \{S\}\_\mathcal \{X\} \]
are measured considering the Finsler metric which consists of the norm of \[\mathcal \{X\}\]
at each tangent space of \[\mathcal \{S\}\_\mathcal \{X\} \]
. The initial value problem is solved, for the case when \[\mathcal \{A\}\]
is a von Neumann algebra and \[\mathcal \{X\}\]
is selfdual: for any element x 0 ∈ \[\mathcal \{S\}\_\mathcal \{X\} \]
and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ \[\mathcal \{L\}\_\mathcal \{A\} (\mathcal \{X\})\]
, Z* = −Z and ∥Z∥ ≤ π, such that γ(0) = x 0 and \[\dot\{\gamma \}\]
(0) = ν, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x 0, x 1 ∈ \[\mathcal \{S\}\_\mathcal \{X\} \]
, find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denote by f 0 the selfadjoint projection I − x 0 ⊗ x 0, if the algebra f 0 \[\mathcal \{L\}\_\mathcal \{A\} (\mathcal \{X\})\]
f 0 is finite dimensional, then there exists a curve γ joining x 0 and x 1, which is minimizing along its path.},

author = {Esteban Andruchow, Alejandro Varela},

journal = {Open Mathematics},

keywords = {C*-modules; spheres; geodesics; -modules},

language = {eng},

number = {4},

pages = {639-653},

title = {Metrics in the sphere of a C*-module},

url = {http://eudml.org/doc/269531},

volume = {5},

year = {2007},

}

TY - JOUR

AU - Esteban Andruchow

AU - Alejandro Varela

TI - Metrics in the sphere of a C*-module

JO - Open Mathematics

PY - 2007

VL - 5

IS - 4

SP - 639

EP - 653

AB - Given a unital C*-algebra \[\mathcal {A}\]
and a right C*-module \[\mathcal {X}\]
over \[\mathcal {A}\]
, we consider the problem of finding short smooth curves in the sphere \[\mathcal {S}_\mathcal {X} \]
= x ∈ \[\mathcal {X}\]
: 〈x, x〉 = 1. Curves in \[\mathcal {S}_\mathcal {X} \]
are measured considering the Finsler metric which consists of the norm of \[\mathcal {X}\]
at each tangent space of \[\mathcal {S}_\mathcal {X} \]
. The initial value problem is solved, for the case when \[\mathcal {A}\]
is a von Neumann algebra and \[\mathcal {X}\]
is selfdual: for any element x 0 ∈ \[\mathcal {S}_\mathcal {X} \]
and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ \[\mathcal {L}_\mathcal {A} (\mathcal {X})\]
, Z* = −Z and ∥Z∥ ≤ π, such that γ(0) = x 0 and \[\dot{\gamma }\]
(0) = ν, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x 0, x 1 ∈ \[\mathcal {S}_\mathcal {X} \]
, find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denote by f 0 the selfadjoint projection I − x 0 ⊗ x 0, if the algebra f 0 \[\mathcal {L}_\mathcal {A} (\mathcal {X})\]
f 0 is finite dimensional, then there exists a curve γ joining x 0 and x 1, which is minimizing along its path.

LA - eng

KW - C*-modules; spheres; geodesics; -modules

UR - http://eudml.org/doc/269531

ER -

## References

top- [1] E. Andruchow, G. Corach and M. Mbekhta: “On the geometry of generalized inverses”, Math. Nachr., Vol. 278, (2005), no. 7–8, pp. 756–770. http://dx.doi.org/10.1002/mana.200310270 Zbl1086.46037
- [2] E. Andruchow, G. Corach and D. Stojanoff: “Geometry of the sphere of a Hilbert module”, Math. Proc. Cambridge Philos. Soc., Vol. 127, (1999), no. 2, pp. 295–315. http://dx.doi.org/10.1017/S0305004199003771 Zbl0945.46042
- [3] E. Andruchow, G. Corach and D. Stojanoff: “Projective spaces of a C*-algebra”, Integral Equations Operator Theory, Vol. 37, (2000), no. 2, pp. 143–168. http://dx.doi.org/10.1007/BF01192421 Zbl0962.46040
- [4] E. Andruchow and A. Varela: “C*-modular vector states”, Integral Equations Operator Theory, Vol. 52, (2005), pp. 149–163. http://dx.doi.org/10.1007/s00020-002-1280-y Zbl1098.46047
- [5] C.J. Atkin: “The Finsler geometry of groups of isometries of Hilbert space”, J. Austral. Math. Soc. Ser. A, Vol. 42, (1987), pp. 196–222. http://dx.doi.org/10.1017/S1446788700028202 Zbl0614.58007
- [6] C. Davis, W.M. Kahan and H.F. Weinberger: “Norm preserving dilations and their applications to optimal error bounds”, SIAM J. Numer. Anal., Vol. 19, (1982), pp. 445–469. http://dx.doi.org/10.1137/0719029 Zbl0491.47003
- [7] C.E. Durán, L.E. Mata-Lorenzo and L. Recht: “Metric geometry in homogeneous spaces of the unitary group of a C*-algebra. Part I. Minimal curves”, Adv. Math., Vol. 184, (2004), no. 2, pp. 342–366. http://dx.doi.org/10.1016/S0001-8708(03)00148-8 Zbl1060.53076
- [8] C.E. Durán, L.E. Mata-Lorenzo and L. Recht: “Metric geometry in homogeneous spaces of the unitary group of a C*-algebra. Part II. Geodesics joining fixed endpoints”, Integral Equations Operator Theory, Vol. 53, (2005), no. 1, pp. 33–50. http://dx.doi.org/10.1007/s00020-003-1305-1 Zbl1096.53044
- [9] S. Kobayashi and K. Nomizu: Foundations of differential geometry, Vol. II. Reprint of the 1969 original, Wiley Classics Library, John Wiley & Sons, New York, 1996. Zbl0119.37502
- [10] M.G. Krein: “The theory of selfadjoint extensions of semibounded Hermitian transformations and its applications”, Mat. Sb., Vol. 20, (1947), pp. 431–495, Vol. 21, (1947), pp. 365–404 (in Russian). Zbl0029.14103
- [11] E.C. Lance: “Hilbert C*-modules, A toolkit for operator algebraists”, London Math. Soc. Lecture Note Ser., Vol. 210, Cambridge University Press, Cambridge, 1995. Zbl0822.46080
- [12] P.R. Halmos and J.E. McLaughlin: “Partial isometries”, Pacific J. Math., Vol. 13, (1963), pp. 585–596. Zbl0189.13402
- [13] L.E. Mata-Lorenzo and L. Recht: “Infinite-dimensional homogeneous reductive spaces”, Acta Cient. Venezolana, Vol. 43, (1992), pp. 76–90. Zbl0765.53038
- [14] S. Parrott: “On a quotient norm and the Sz.-Nagy-Foias lifting theorem”, J. Funct. Anal., Vol. 30, (1978), no. 3, pp. 311–328. http://dx.doi.org/10.1016/0022-1236(78)90060-5 Zbl0409.47004
- [15] W.L. Paschke: “Inner product modules over B*-algebras”, Trans. Amer. Math. Soc., Vol. 182, (1973), pp. 443–468. http://dx.doi.org/10.2307/1996542 Zbl0239.46062
- [16] F. Riesz and B. Sz.-Nagy: Functional Analysis, Frederick Ungar Publishing Co., New York, 1955.

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